g10-12-02 · Additional mathematics
The smallest shared multiple
Why is multiplying A and B usually excessive?
1 MIN · PREDICT ON PAPER
Compare all factor copies in AB with the copies needed by a common multiple.
Discuss or write first. No typed answer is required to continue.
Monic result: LCM = (x−1)^2(x+1)^3
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
A common multiple must contain enough copies for both source polynomials, so each exponent is the maximum, not the sum. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: All polynomials are monic and nonzero, with only the displayed distinct factors.
Mathematical justification: A common-multiple exponent is at least both source exponents, hence at least their maximum. The construction attains it; min(u,v)+max(u,v)=u+v proves the product identity.
1 MIN · EXIT CHECK
Find the monic LCM of x−1 and (x−1)².
Check after answering
(x−1)², not (x−1)³.
Teacher guidance and extension
Before: GCD and factor counts
Ask: Ask which factor can be removed without losing either divisibility requirement.
Watch for: A multiple need not include two separate copies of a factor already shared.
Challenge: Verify GCD(A,B)·LCM(A,B)=AB factor by factor.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
Paper responses are enough. Drafts remain in this tab; choose local saving to include them in the existing complete backup.
Complete backup and restore